*3.2. Introducing Linear Eigenvalue Equations*

The stability of the steady flow solutions can be explored by setting *f*(*η*) = *f*0(*η*), *h*(*η*) = *h*0(*η*) and *<sup>θ</sup>*(*η*) = *<sup>θ</sup>*0(*η*), where it satisfied the boundary value problems (11)–(13) and (15). Thus, the following equations are introduced (Weidman et al. [63]):

$$f(\eta,\tau) = f\_0(\eta) + e^{-\gamma\tau} F(\eta,\tau), \; h(\eta,\tau) = h\_0(\eta) + e^{-\gamma\tau} H(\eta,\tau), \; \theta(\eta,\tau) = \theta\_0(\eta) + e^{-\gamma\tau} G(\eta,\tau), \tag{24}$$

where *<sup>F</sup>*(*η*, *<sup>τ</sup>*), *<sup>H</sup>*(*η*, *<sup>τ</sup>*), *<sup>G</sup>*(*η*, *τ*) and their derivatives are small then *f*0(*η*), *h*0(*η*) and *<sup>θ</sup>*0(*η*). In addition, *γ* is the unknown eigenvalue which will be used to specify the stability of the solutions. Substitute Equation (24) into (20)–(22) and let *τ* → 0, in which *<sup>F</sup>*(*η*) = *<sup>F</sup>*0(*η*), *<sup>H</sup>*(*η*) = *<sup>H</sup>*0(*η*) and *<sup>G</sup>*(*η*) = *<sup>G</sup>*0(*η*), thereby the linearized eigenvalue equations relevant to the problem are

$$\frac{\mu\_{\text{hfr}}/\mu\_f}{\rho\_{\text{hfr}}/\rho\_f}(1+K)F\_0^{\prime\prime\prime} + \left(f\_0 + \frac{A}{2}\eta\right)F\_0^{\prime\prime} + F\_0f\_0^{\prime\prime} - \left(2f\_0^{\prime} + A - \gamma\right)F\_0^{\prime} + \frac{K}{\rho\_{\text{hfr}}/\rho\_f}H\_0^{\prime} = 0\tag{25}$$

$$\frac{1}{\rho\_{\rm mf}/\rho\_f} \left( \frac{\mu\_{\rm mf}}{\mu\_f} + \frac{K}{2} \right) H\_0'' + \left( f\_0 - \frac{A}{2} \eta \right) H\_0' + F\_0 h\_0' - F\_0' h\_0 - \left( f\_0' + \frac{3}{2} A - \gamma \right) H\_0 - \frac{K}{\rho\_{\rm mf}/\rho\_f} \left( 2H\_0 + F\_0'' \right) = 0 \tag{26}$$

$$\frac{1}{2\left(\rho\mathbb{C}\_{p}\right)\_{\text{hul}}/\left(\rho\mathbb{C}\_{p}\right)\_{f}}\left(\frac{k\_{\text{hul}}}{k\_{f}}+\frac{4}{3}\text{R}d\right)\mathbb{G}\_{0}\prime\prime+\left(f\_{0}-\frac{A}{2}\eta\right)\mathbb{G}\_{0}\prime+F\_{0}\mathbb{A}\_{0}\prime-2F\_{0}\prime\theta\_{0}-\left(2f\_{0}\prime+\frac{3}{2}A-\gamma\right)\mathbb{G}\_{0} = 0\tag{27}$$

The conditions now take the following form

$$\begin{array}{ll} \text{Fo}'(0) = 0, \; \text{Fo}(0) = 0, \; \text{H}\_{0}(0) = -n \, \text{Fo}''(0), \; \text{G}\_{0}(0) = 0, \\\ \text{F}\_{0}'(\eta) \to 0, \; \text{H}\_{0}(\eta) \to 0, \; \text{G}\_{0}(\eta) \to 0, \; \text{as} \, \eta \to \infty \end{array} \tag{28}$$
